Topology

Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. When the discipline was first introduced it was called analysis situs (Latin analysis of place).

Some theorems in general topology

• Every closed interval in R of finite length is compact. More is true: In Rⁿ, a set is compact iff it is closed and bounded. (See Heine-Borel theorem).
• Every continuous image of a compact space is compact.
• Tychonoff's theorem: The (arbitrary) product of compact spaces is compact.
• A compact subspace of a Hausdorff space is closed.
• Every sequence of points in a compact metric space has a convergent subsequence.
• Every interval in R is connected.
• The continuous image of a connected space is connected.
• A metric space is Hausdorff, also normal and paracompact.
• The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
• The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
• The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
• On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
• Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.
• Topological spaces have a topological dimension.